  
  [1X56 [33X[0;0YRings[133X[101X
  
  [33X[0;0YThis   chapter   deals   with   domains   that   are  additive  groups  (see
  [2XIsAdditiveGroup[102X  ([14X55.1-6[114X) closed under multiplication [10X*[110X. Such a domain, if [10X*[110X
  and  [10X+[110X  are distributive, is called a [13Xring[113X in [5XGAP[105X. Each division ring, field
  (see [14X58[114X), or algebra (see [14X62[114X) is a ring. Important examples of rings are the
  integers (see [14X14[114X) and matrix rings.[133X
  
  [33X[0;0YIn  the  case  of  a  [13Xring-with-one[113X,  additional multiplicative structure is
  present,  see [2XIsRingWithOne[102X  ([14X56.3-1[114X).  There is a little support in [5XGAP[105X for
  rings  that  have  no  additional  structure: it is possible to perform some
  computations for small finite rings; infinite rings are handled by [5XGAP[105X in an
  acceptable way in the case that they are algebras.[133X
  
  [33X[0;0YAlso,  the  [5XSONATA[105X  package  provides  support for near-rings, and a related
  functionality  for  multiplicative  semigroups of near-rings is available in
  the [5XSmallsemi[105X package.[133X
  
  [33X[0;0YSeveral  functions  for  ring elements, such as [2XIsPrime[102X ([14X56.5-8[114X) and [2XFactors[102X
  ([14X56.5-9[114X),  are defined only relative to a ring [3XR[103X, which can be entered as an
  optional  argument;  if  [3XR[103X is omitted then a [13Xdefault ring[113X is formed from the
  ring elements given as arguments, see [2XDefaultRing[102X ([14X56.1-3[114X).[133X
  
  
  [1X56.1 [33X[0;0YGenerating Rings[133X[101X
  
  [1X56.1-1 IsRing[101X
  
  [29X[2XIsRing[102X( [3XR[103X ) [32X property
  
  [33X[0;0YA  [13Xring[113X  in  [5XGAP[105X is an additive group (see [2XIsAdditiveGroup[102X ([14X55.1-6[114X)) that is
  also a magma (see [2XIsMagma[102X ([14X35.1-1[114X)), such that addition [10X+[110X and multiplication
  [10X*[110X are distributive, see [2XIsDistributive[102X ([14X56.4-5[114X).[133X
  
  [33X[0;0YThe multiplication need [13Xnot[113X be associative (see [2XIsAssociative[102X ([14X35.4-7[114X)). For
  example, a Lie algebra (see [14X64[114X) is regarded as a ring in [5XGAP[105X.[133X
  
  
  [1X56.1-2 [33X[0;0YRing[133X[101X
  
  [29X[2XRing[102X( [3Xr[103X, [3Xs[103X, [3X...[103X ) [32X function
  [29X[2XRing[102X( [3Xcoll[103X ) [32X function
  
  [33X[0;0YIn  the  first  form  [2XRing[102X  returns  the smallest ring that contains all the
  elements  [3Xr[103X,  [3Xs[103X,  [22X...[122X In the second form [2XRing[102X returns the smallest ring that
  contains  all  the elements in the collection [3Xcoll[103X. If any element is not an
  element  of  a  ring  or  if  the elements lie in no common ring an error is
  raised.[133X
  
  [33X[0;0Y[2XRing[102X  differs from [2XDefaultRing[102X ([14X56.1-3[114X) in that it returns the smallest ring
  in  which  the  elements lie, while [2XDefaultRing[102X ([14X56.1-3[114X) may return a larger
  ring if that makes sense.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRing( 2, E(4) );[127X[104X
    [4X[28X<ring with 2 generators>[128X[104X
  [4X[32X[104X
  
  
  [1X56.1-3 [33X[0;0YDefaultRing[133X[101X
  
  [29X[2XDefaultRing[102X( [3Xr[103X, [3Xs[103X, [3X...[103X ) [32X function
  [29X[2XDefaultRing[102X( [3Xcoll[103X ) [32X function
  
  [33X[0;0YIn  the first form [2XDefaultRing[102X returns a ring that contains all the elements
  [3Xr[103X,  [3Xs[103X,  [22X...[122X etc. In the second form [2XDefaultRing[102X returns a ring that contains
  all the elements in the collection [3Xcoll[103X. If any element is not an element of
  a ring or if the elements lie in no common ring an error is raised.[133X
  
  [33X[0;0YThe  ring returned by [2XDefaultRing[102X need not be the smallest ring in which the
  elements  lie.  For example for elements from cyclotomic fields, [2XDefaultRing[102X
  may  return  the  ring of integers of the smallest cyclotomic field in which
  the  elements  lie, which need not be the smallest ring overall, because the
  elements  may  in  fact  lie in a smaller number field which is itself not a
  cyclotomic field.[133X
  
  [33X[0;0Y(For the exact definition of the default ring of a certain type of elements,
  look at the corresponding method installation.)[133X
  
  [33X[0;0Y[2XDefaultRing[102X  is  used  by  ring functions such as [2XQuotient[102X ([14X56.1-9[114X), [2XIsPrime[102X
  ([14X56.5-8[114X), [2XFactors[102X ([14X56.5-9[114X), or [2XGcd[102X ([14X56.7-1[114X) if no explicit ring is given.[133X
  
  [33X[0;0Y[2XRing[102X  ([14X56.1-2[114X) differs from [2XDefaultRing[102X in that it returns the smallest ring
  in  which  the  elements  lie, while [2XDefaultRing[102X may return a larger ring if
  that makes sense.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDefaultRing( 2, E(4) );[127X[104X
    [4X[28XGaussianIntegers[128X[104X
  [4X[32X[104X
  
  [1X56.1-4 RingByGenerators[101X
  
  [29X[2XRingByGenerators[102X( [3XC[103X ) [32X operation
  
  [33X[0;0Y[2XRingByGenerators[102X   returns  the  ring  generated  by  the  elements  in  the
  collection  [3XC[103X,  i. e.,  the closure of [3XC[103X under addition, multiplication, and
  taking additive inverses.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRingByGenerators([ 2, E(4) ]);[127X[104X
    [4X[28X<ring with 2 generators>[128X[104X
  [4X[32X[104X
  
  [1X56.1-5 DefaultRingByGenerators[101X
  
  [29X[2XDefaultRingByGenerators[102X( [3Xcoll[103X ) [32X operation
  
  [33X[0;0YFor a collection [3Xcoll[103X, returns a default ring in which [3Xcoll[103X is contained.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDefaultRingByGenerators([ 2, E(4) ]);[127X[104X
    [4X[28XGaussianIntegers[128X[104X
  [4X[32X[104X
  
  [1X56.1-6 GeneratorsOfRing[101X
  
  [29X[2XGeneratorsOfRing[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0Y[2XGeneratorsOfRing[102X  returns  a  list  of  elements such that the ring [3XR[103X is the
  closure  of  these  elements  under  addition,  multiplication,  and  taking
  additive inverses.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=Ring( 2, 1/2 );[127X[104X
    [4X[28X<ring with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfRing( R );[127X[104X
    [4X[28X[ 2, 1/2 ][128X[104X
  [4X[32X[104X
  
  [1X56.1-7 Subring[101X
  
  [29X[2XSubring[102X( [3XR[103X, [3Xgens[103X ) [32X function
  [29X[2XSubringNC[102X( [3XR[103X, [3Xgens[103X ) [32X function
  
  [33X[0;0Yreturns  the  ring with parent [3XR[103X generated by the elements in [3Xgens[103X. When the
  second  form,  [2XSubringNC[102X  is used, it is [13Xnot[113X checked whether all elements in
  [3Xgens[103X lie in [3XR[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:= Integers;[127X[104X
    [4X[28XIntegers[128X[104X
    [4X[25Xgap>[125X [27XS:= Subring( R, [ 4, 6 ] );[127X[104X
    [4X[28X<ring with 1 generators>[128X[104X
    [4X[25Xgap>[125X [27XParent( S );[127X[104X
    [4X[28XIntegers[128X[104X
  [4X[32X[104X
  
  
  [1X56.1-8 [33X[0;0YClosureRing[133X[101X
  
  [29X[2XClosureRing[102X( [3XR[103X, [3Xr[103X ) [32X operation
  [29X[2XClosureRing[102X( [3XR[103X, [3XS[103X ) [32X operation
  
  [33X[0;0YFor  a  ring  [3XR[103X  and either an element [3Xr[103X of its elements family or a ring [3XS[103X,
  [2XClosureRing[102X returns the ring generated by both arguments.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XClosureRing( Integers, E(4) );[127X[104X
    [4X[28X<ring-with-one, with 2 generators>[128X[104X
  [4X[32X[104X
  
  [1X56.1-9 Quotient[101X
  
  [29X[2XQuotient[102X( [[3XR[103X, ][3Xr[103X, [3Xs[103X ) [32X operation
  
  [33X[0;0Y[2XQuotient[102X  returns  the quotient of the two ring elements [3Xr[103X and [3Xs[103X in the ring
  [3XR[103X, if given, and otherwise in their default ring (see [2XDefaultRing[102X ([14X56.1-3[114X)).
  It returns [9Xfail[109X if the quotient does not exist in the respective ring.[133X
  
  [33X[0;0Y(To  perform  the division in the quotient field of a ring, use the quotient
  operator [10X/[110X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQuotient( 2, 3 );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XQuotient( 6, 3 );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X56.2 [33X[0;0YIdeals in Rings[133X[101X
  
  [33X[0;0YA  [13Xleft  ideal[113X  in  a  ring  [22XR[122X  is  a  subring  of  [22XR[122X  that  is closed under
  multiplication with elements of [22XR[122X from the left.[133X
  
  [33X[0;0YA  [13Xright  ideal[113X  in  a  ring  [22XR[122X  is  a  subring  of  [22XR[122X  that is closed under
  multiplication with elements of [22XR[122X from the right.[133X
  
  [33X[0;0YA  [13Xtwo-sided  ideal[113X  or  simply [13Xideal[113X in a ring [22XR[122X is both a left ideal and a
  right ideal in [22XR[122X.[133X
  
  [33X[0;0YSo  being  a  (left/right/two-sided) ideal is not a property of a domain but
  refers  to  the acting ring(s). Hence we must ask, e. g., [10XIsIdeal( [110X[22XR, I[122X[10X )[110X if
  we  want  to know whether the ring [22XI[122X is an ideal in the ring [22XR[122X. The property
  [2XIsTwoSidedIdealInParent[102X  ([14X56.2-3[114X)  can be used to store whether a ring is an
  ideal in its parent.[133X
  
  [33X[0;0Y(Whenever  the  term  [10X"Ideal"[110X  occurs  in an identifier without a specifying
  prefix   [10X"Left"[110X   or  [10X"Right"[110X,  this  means  the  same  as  [10X"TwoSidedIdeal"[110X.
  Conversely,   any  occurrence  of  [10X"TwoSidedIdeal"[110X  can  be  substituted  by
  [10X"Ideal"[110X.)[133X
  
  [33X[0;0YFor  any  of  the  above  kinds  of ideals, there is a notion of generators,
  namely  [2XGeneratorsOfLeftIdeal[102X ([14X56.2-8[114X), [2XGeneratorsOfRightIdeal[102X ([14X56.2-9[114X), and
  [2XGeneratorsOfTwoSidedIdeal[102X  ([14X56.2-7[114X).  The  acting  rings  can be accessed as
  [2XLeftActingRingOfIdeal[102X   ([14X56.2-10[114X)   and   [2XRightActingRingOfIdeal[102X  ([14X56.2-10[114X),
  respectively.  Note  that  ideals  are  detected  from known values of these
  attributes,  especially it is assumed that whenever a domain has both a left
  and a right acting ring then these two are equal.[133X
  
  [33X[0;0YNote  that  we  cannot  use [2XLeftActingDomain[102X ([14X57.1-11[114X) and [10XRightActingDomain[110X
  here,  since  ideals  in  algebras  are themselves vector spaces, and such a
  space  can of course also be a module for an action from the right. In order
  to  make  the  usual  vector space functionality automatically available for
  ideals,  we have to distinguish the left and right module structure from the
  additional closure properties of the ideal.[133X
  
  [33X[0;0YFurther  note  that the attributes denoting ideal generators and acting ring
  are  used  to  create  ideals  if  this  is explicitly wanted, but the ideal
  relation  in  the sense of [2XIsTwoSidedIdeal[102X ([14X56.2-3[114X) is of course independent
  of the presence of the attribute values.[133X
  
  [33X[0;0YIdeals   are  constructed  with  [2XLeftIdeal[102X  ([14X56.2-1[114X),  [2XRightIdeal[102X  ([14X56.2-1[114X),
  [2XTwoSidedIdeal[102X ([14X56.2-1[114X). Principal ideals of the form [22Xx * R[122X, [22XR * x[122X, [22XR * x * R[122X
  can also be constructed with a simple multiplication.[133X
  
  [33X[0;0YCurrently  many methods for dealing with ideals need linear algebra to work,
  so they are mainly applicable to ideals in algebras.[133X
  
  [1X56.2-1 TwoSidedIdeal[101X
  
  [29X[2XTwoSidedIdeal[102X( [3XR[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function
  [29X[2XIdeal[102X( [3XR[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function
  [29X[2XLeftIdeal[102X( [3XR[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function
  [29X[2XRightIdeal[102X( [3XR[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function
  
  [33X[0;0YLet  [3XR[103X  be  a  ring,  and  [3Xgens[103X  a  list  of  collection  of  elements in [3XR[103X.
  [2XTwoSidedIdeal[102X,  [2XLeftIdeal[102X,  and  [2XRightIdeal[102X  return  the two-sided, left, or
  right  ideal, respectively, [22XI[122X in [3XR[103X that is generated by [3Xgens[103X. The ring [3XR[103X can
  be  accessed  as  [2XLeftActingRingOfIdeal[102X  ([14X56.2-10[114X) or [2XRightActingRingOfIdeal[102X
  ([14X56.2-10[114X) (or both) of [22XI[122X.[133X
  
  [33X[0;0YIf  [3XR[103X  is  a left [22XF[122X-module then also [22XI[122X is a left [22XF[122X-module, in particular the
  [2XLeftActingDomain[102X ([14X57.1-11[114X) values of [3XR[103X and [22XI[122X are equal.[133X
  
  [33X[0;0YIf the optional argument [10X"basis"[110X is given then [3Xgens[103X are assumed to be a list
  of  basis vectors of [22XI[122X viewed as a free [22XF[122X-module. (This is mainly applicable
  to  ideals in algebras.) In this case, it is [13Xnot[113X checked whether [3Xgens[103X really
  is linearly independent and whether [3Xgens[103X is a subset of [3XR[103X.[133X
  
  [33X[0;0Y[2XIdeal[102X is simply a synonym of [2XTwoSidedIdeal[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:= Integers;;[127X[104X
    [4X[25Xgap>[125X [27XI:= Ideal( R, [ 2 ] );[127X[104X
    [4X[28X<two-sided ideal in Integers, (1 generators)>[128X[104X
  [4X[32X[104X
  
  [1X56.2-2 TwoSidedIdealNC[101X
  
  [29X[2XTwoSidedIdealNC[102X( [3XR[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function
  [29X[2XIdealNC[102X( [3XR[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function
  [29X[2XLeftIdealNC[102X( [3XR[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function
  [29X[2XRightIdealNC[102X( [3XR[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function
  
  [33X[0;0YThe  effects  of [2XTwoSidedIdealNC[102X, [2XLeftIdealNC[102X, and [2XRightIdealNC[102X are the same
  as  [2XTwoSidedIdeal[102X  ([14X56.2-1[114X),  [2XLeftIdeal[102X  ([14X56.2-1[114X),  and [2XRightIdeal[102X ([14X56.2-1[114X),
  respectively, but they do not check whether all entries of [3Xgens[103X lie in [3XR[103X.[133X
  
  [1X56.2-3 IsTwoSidedIdeal[101X
  
  [29X[2XIsTwoSidedIdeal[102X( [3XR[103X, [3XI[103X ) [32X operation
  [29X[2XIsLeftIdeal[102X( [3XR[103X, [3XI[103X ) [32X operation
  [29X[2XIsRightIdeal[102X( [3XR[103X, [3XI[103X ) [32X operation
  [29X[2XIsTwoSidedIdealInParent[102X( [3XI[103X ) [32X property
  [29X[2XIsLeftIdealInParent[102X( [3XI[103X ) [32X property
  [29X[2XIsRightIdealInParent[102X( [3XI[103X ) [32X property
  
  [33X[0;0YThe  properties  [2XIsTwoSidedIdealInParent[102X  etc., are attributes of the ideal,
  and once known they are stored in the ideal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:= FullMatrixAlgebra( Rationals, 3 );[127X[104X
    [4X[28X( Rationals^[ 3, 3 ] )[128X[104X
    [4X[25Xgap>[125X [27XI:= Ideal( A, [ Random( A ) ] );[127X[104X
    [4X[28X<two-sided ideal in ( Rationals^[ 3, 3 ] ), (1 generators)>[128X[104X
    [4X[25Xgap>[125X [27XIsTwoSidedIdeal( A, I );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X56.2-4 TwoSidedIdealByGenerators[101X
  
  [29X[2XTwoSidedIdealByGenerators[102X( [3XR[103X, [3Xgens[103X ) [32X operation
  [29X[2XIdealByGenerators[102X( [3XR[103X, [3Xgens[103X ) [32X operation
  
  [33X[0;0Y[2XTwoSidedIdealByGenerators[102X returns the ring that is generated by the elements
  of  the  collection  [3Xgens[103X under addition, multiplication, and multiplication
  with elements of the ring [3XR[103X from the left and from the right.[133X
  
  [33X[0;0Y[3XR[103X    can    be    accessed    by    [2XLeftActingRingOfIdeal[102X    ([14X56.2-10[114X)    or
  [2XRightActingRingOfIdeal[102X     ([14X56.2-10[114X),    [3Xgens[103X    can    be    accessed    by
  [2XGeneratorsOfTwoSidedIdeal[102X ([14X56.2-7[114X).[133X
  
  [1X56.2-5 LeftIdealByGenerators[101X
  
  [29X[2XLeftIdealByGenerators[102X( [3XR[103X, [3Xgens[103X ) [32X operation
  
  [33X[0;0Y[2XLeftIdealByGenerators[102X  returns the ring that is generated by the elements of
  the  collection [3Xgens[103X under addition, multiplication, and multiplication with
  elements of the ring [3XR[103X from the left.[133X
  
  [33X[0;0Y[3XR[103X  can  be accessed by [2XLeftActingRingOfIdeal[102X ([14X56.2-10[114X), [3Xgens[103X can be accessed
  by [2XGeneratorsOfLeftIdeal[102X ([14X56.2-8[114X).[133X
  
  [1X56.2-6 RightIdealByGenerators[101X
  
  [29X[2XRightIdealByGenerators[102X( [3XR[103X, [3Xgens[103X ) [32X operation
  
  [33X[0;0Y[2XRightIdealByGenerators[102X returns the ring that is generated by the elements of
  the  collection [3Xgens[103X under addition, multiplication, and multiplication with
  elements of the ring [3XR[103X from the right.[133X
  
  [33X[0;0Y[3XR[103X  can be accessed by [2XRightActingRingOfIdeal[102X ([14X56.2-10[114X), [3Xgens[103X can be accessed
  by [2XGeneratorsOfRightIdeal[102X ([14X56.2-9[114X).[133X
  
  [1X56.2-7 GeneratorsOfTwoSidedIdeal[101X
  
  [29X[2XGeneratorsOfTwoSidedIdeal[102X( [3XI[103X ) [32X attribute
  [29X[2XGeneratorsOfIdeal[102X( [3XI[103X ) [32X attribute
  
  [33X[0;0Yis  a  list of generators for the ideal [3XI[103X, with respect to the action of the
  rings  that  are stored as the values of [2XLeftActingRingOfIdeal[102X ([14X56.2-10[114X) and
  [2XRightActingRingOfIdeal[102X   ([14X56.2-10[114X),  from  the  left  and  from  the  right,
  respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:= FullMatrixAlgebra( Rationals, 3 );;[127X[104X
    [4X[25Xgap>[125X [27XI:= Ideal( A, [ One( A ) ] );;[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfIdeal( I );[127X[104X
    [4X[28X[ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ][128X[104X
  [4X[32X[104X
  
  [1X56.2-8 GeneratorsOfLeftIdeal[101X
  
  [29X[2XGeneratorsOfLeftIdeal[102X( [3XI[103X ) [32X attribute
  
  [33X[0;0Yis  a  list  of  generators for the left ideal [3XI[103X, with respect to the action
  from   the   left   of   the   ring   that   is   stored  as  the  value  of
  [2XLeftActingRingOfIdeal[102X ([14X56.2-10[114X).[133X
  
  [1X56.2-9 GeneratorsOfRightIdeal[101X
  
  [29X[2XGeneratorsOfRightIdeal[102X( [3XI[103X ) [32X attribute
  
  [33X[0;0Yis  a  list  of generators for the right ideal [3XI[103X, with respect to the action
  from   the   right   of   the   ring   that   is  stored  as  the  value  of
  [2XRightActingRingOfIdeal[102X ([14X56.2-10[114X).[133X
  
  [1X56.2-10 LeftActingRingOfIdeal[101X
  
  [29X[2XLeftActingRingOfIdeal[102X( [3XI[103X ) [32X attribute
  [29X[2XRightActingRingOfIdeal[102X( [3XI[103X ) [32X attribute
  
  [33X[0;0Yreturns the left (resp. right) acting ring of an ideal [3XI[103X.[133X
  
  [1X56.2-11 AsLeftIdeal[101X
  
  [29X[2XAsLeftIdeal[102X( [3XR[103X, [3XS[103X ) [32X operation
  [29X[2XAsRightIdeal[102X( [3XR[103X, [3XS[103X ) [32X operation
  [29X[2XAsTwoSidedIdeal[102X( [3XR[103X, [3XS[103X ) [32X operation
  
  [33X[0;0YLet [3XS[103X be a subring of the ring [3XR[103X.[133X
  
  [33X[0;0YIf  [3XS[103X  is  a  left  ideal  in  [3XR[103X  then  [2XAsLeftIdeal[102X returns this left ideal,
  otherwise [9Xfail[109X is returned.[133X
  
  [33X[0;0YIf  [3XS[103X  is  a  right  ideal  in [3XR[103X then [2XAsRightIdeal[102X returns this right ideal,
  otherwise [9Xfail[109X is returned.[133X
  
  [33X[0;0YIf  [3XS[103X  is a two-sided ideal in [3XR[103X then [2XAsTwoSidedIdeal[102X returns this two-sided
  ideal, otherwise [9Xfail[109X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:= FullMatrixAlgebra( Rationals, 3 );;[127X[104X
    [4X[25Xgap>[125X [27XB:= DirectSumOfAlgebras( A, A );[127X[104X
    [4X[28X<algebra over Rationals, with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XC:= Subalgebra( B, Basis( B ){[1..9]} );[127X[104X
    [4X[28X<algebra over Rationals, with 9 generators>[128X[104X
    [4X[25Xgap>[125X [27XI:= AsTwoSidedIdeal( B, C );[127X[104X
    [4X[28X<two-sided ideal in <algebra of dimension 18 over Rationals>, [128X[104X
    [4X[28X  (9 generators)>[128X[104X
  [4X[32X[104X
  
  
  [1X56.3 [33X[0;0YRings With One[133X[101X
  
  [1X56.3-1 IsRingWithOne[101X
  
  [29X[2XIsRingWithOne[102X( [3XR[103X ) [32X property
  
  [33X[0;0YA  [13Xring-with-one[113X  in  [5XGAP[105X  is  a  ring  (see [2XIsRing[102X ([14X56.1-1[114X)) that is also a
  magma-with-one (see [2XIsMagmaWithOne[102X ([14X35.1-2[114X)).[133X
  
  [33X[0;0YNote that the identity and the zero of a ring-with-one need [13Xnot[113X be distinct.
  This  means  that  a  ring  that  consists  only  of its zero element can be
  regarded as a ring-with-one.[133X
  
  [33X[0;0YThis  is  especially  useful in the case of finitely presented rings, in the
  sense that each factor of a ring-with-one is again a ring-with-one.[133X
  
  
  [1X56.3-2 [33X[0;0YRingWithOne[133X[101X
  
  [29X[2XRingWithOne[102X( [3Xr[103X, [3Xs[103X, [3X...[103X ) [32X function
  [29X[2XRingWithOne[102X( [3Xcoll[103X ) [32X function
  
  [33X[0;0YIn  the  first  form  [2XRingWithOne[102X  returns  the  smallest ring with one that
  contains  all  the elements [3Xr[103X, [3Xs[103X, [22X...[122X In the second form [2XRingWithOne[102X returns
  the  smallest ring with one that contains all the elements in the collection
  [3XC[103X.  If  any element is not an element of a ring or if the elements lie in no
  common ring an error is raised.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRingWithOne( [ 4, 6 ] );[127X[104X
    [4X[28XIntegers[128X[104X
  [4X[32X[104X
  
  [1X56.3-3 RingWithOneByGenerators[101X
  
  [29X[2XRingWithOneByGenerators[102X( [3Xcoll[103X ) [32X operation
  
  [33X[0;0Y[2XRingWithOneByGenerators[102X  returns the ring-with-one generated by the elements
  in  the  collection  [3Xcoll[103X,  i. e.,  the  closure  of  [3Xcoll[103X  under  addition,
  multiplication,  taking  additive  inverses,  and  taking the identity of an
  element.[133X
  
  [1X56.3-4 GeneratorsOfRingWithOne[101X
  
  [29X[2XGeneratorsOfRingWithOne[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0Y[2XGeneratorsOfRingWithOne[102X  returns  a list of elements such that the ring [3XR[103X is
  the  closure  of  these  elements  under  addition,  multiplication,  taking
  additive inverses, and taking the identity element [10XOne( [3XR[103X[10X )[110X.[133X
  
  [33X[0;0Y[3XR[103X itself need [13Xnot[113X be known to be a ring-with-one.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:= RingWithOne( [ 4, 6 ] );[127X[104X
    [4X[28XIntegers[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfRingWithOne( R );[127X[104X
    [4X[28X[ 1 ][128X[104X
  [4X[32X[104X
  
  [1X56.3-5 SubringWithOne[101X
  
  [29X[2XSubringWithOne[102X( [3XR[103X, [3Xgens[103X ) [32X function
  [29X[2XSubringWithOneNC[102X( [3XR[103X, [3Xgens[103X ) [32X function
  
  [33X[0;0Yreturns  the  ring with one with parent [3XR[103X generated by the elements in [3Xgens[103X.
  When  the  second  form, [2XSubringWithOneNC[102X is used, it is [13Xnot[113X checked whether
  all elements in [3Xgens[103X lie in [3XR[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:= SubringWithOne( Integers, [ 4, 6 ] );[127X[104X
    [4X[28XIntegers[128X[104X
    [4X[25Xgap>[125X [27XParent( R );[127X[104X
    [4X[28XIntegers[128X[104X
  [4X[32X[104X
  
  
  [1X56.4 [33X[0;0YProperties of Rings[133X[101X
  
  [1X56.4-1 IsIntegralRing[101X
  
  [29X[2XIsIntegralRing[102X( [3XR[103X ) [32X property
  
  [33X[0;0YA  ring-with-one  [3XR[103X is integral if it is commutative, contains no nontrivial
  zero divisors, and if its identity is distinct from its zero.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsIntegralRing( Integers );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X56.4-2 IsUniqueFactorizationRing[101X
  
  [29X[2XIsUniqueFactorizationRing[102X( [3XR[103X ) [32X Category
  
  [33X[0;0YA  ring  [3XR[103X  is  called a [13Xunique factorization ring[113X if it is an integral ring
  (see [2XIsIntegralRing[102X  ([14X56.4-1[114X)),  and  every  nonzero  element  has  a unique
  factorization  into  irreducible  elements, i.e., a unique representation as
  product  of  irreducibles (see [2XIsIrreducibleRingElement[102X ([14X56.5-7[114X)). Unique in
  this  context  means  unique  up  to  permutations  of the factors and up to
  multiplication of the factors by units (see [2XUnits[102X ([14X56.5-2[114X)).[133X
  
  [33X[0;0YMathematically,  a  field  should  therefore  also be a unique factorization
  ring,  since  every  nonzero element is a unit. In [5XGAP[105X, however, at least at
  present  fields  do  not  lie in the filter [2XIsUniqueFactorizationRing[102X, since
  operations   such  as  [2XFactors[102X  ([14X56.5-9[114X),  [2XGcd[102X  ([14X56.7-1[114X),  [2XStandardAssociate[102X
  ([14X56.5-5[114X) and so on do not apply to fields (the results would be trivial, and
  not  especially  useful) and methods which require their arguments to lie in
  [2XIsUniqueFactorizationRing[102X expect these operations to work.[133X
  
  [33X[0;0Y(Note  that  we  cannot  install  a subset maintained method for this filter
  since  the  factorization  of an element needs not exist in a subring. As an
  example,  consider  the subring [22X4 ℕ + 1[122X of the ring [22X4 ℤ + 1[122X; in the subring,
  the  element [22X3 ⋅ 3 ⋅ 11 ⋅ 7[122X has the two factorizations [22X33 ⋅ 21 = 9 ⋅ 77[122X, but
  in  the  large  ring there is the unique factorization [22X(-3) ⋅ (-3) ⋅ (-11) ⋅
  (-7)[122X,  and  it  is  easy  to  see that every element in [22X4 ℤ + 1[122X has a unique
  factorization.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsUniqueFactorizationRing( PolynomialRing( Rationals, 1 ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X56.4-3 IsLDistributive[101X
  
  [29X[2XIsLDistributive[102X( [3XC[103X ) [32X property
  
  [33X[0;0Yis  [9Xtrue[109X if the relation [22Xa * ( b + c ) = ( a * b ) + ( a * c )[122X holds for all
  elements [22Xa[122X, [22Xb[122X, [22Xc[122X in the collection [3XC[103X, and [9Xfalse[109X otherwise.[133X
  
  [1X56.4-4 IsRDistributive[101X
  
  [29X[2XIsRDistributive[102X( [3XC[103X ) [32X property
  
  [33X[0;0Yis  [9Xtrue[109X if the relation [22X( a + b ) * c = ( a * c ) + ( b * c )[122X holds for all
  elements [22Xa[122X, [22Xb[122X, [22Xc[122X in the collection [3XC[103X, and [9Xfalse[109X otherwise.[133X
  
  [1X56.4-5 IsDistributive[101X
  
  [29X[2XIsDistributive[102X( [3XC[103X ) [32X property
  
  [33X[0;0Yis  [9Xtrue[109X  if  the  collection  [3XC[103X  is  both  left and right distributive (see
  [2XIsLDistributive[102X ([14X56.4-3[114X), [2XIsRDistributive[102X ([14X56.4-4[114X)), and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsDistributive( Integers );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X56.4-6 IsAnticommutative[101X
  
  [29X[2XIsAnticommutative[102X( [3XR[103X ) [32X property
  
  [33X[0;0Yis  [9Xtrue[109X  if the relation [22Xa * b = - b * a[122X holds for all elements [22Xa[122X, [22Xb[122X in the
  ring [3XR[103X, and [9Xfalse[109X otherwise.[133X
  
  [1X56.4-7 IsZeroSquaredRing[101X
  
  [29X[2XIsZeroSquaredRing[102X( [3XR[103X ) [32X property
  
  [33X[0;0Yis [9Xtrue[109X if [22Xa * a[122X is the zero element of the ring [3XR[103X for all [22Xa[122X in [3XR[103X, and [9Xfalse[109X
  otherwise.[133X
  
  [1X56.4-8 IsJacobianRing[101X
  
  [29X[2XIsJacobianRing[102X( [3XR[103X ) [32X property
  
  [33X[0;0Yis [9Xtrue[109X if the Jacobi identity holds in the ring [3XR[103X, and [9Xfalse[109X otherwise. The
  Jacobi  identity  means  that [22Xx * (y * z) + z * (x * y) + y * (z * x)[122X is the
  zero element of [3XR[103X, for all elements [22Xx[122X, [22Xy[122X, [22Xz[122X in [3XR[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FullMatrixLieAlgebra( GF( 5 ), 7 );[127X[104X
    [4X[28X<Lie algebra over GF(5), with 13 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsJacobianRing( L );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X56.5 [33X[0;0YUnits and Factorizations[133X[101X
  
  [1X56.5-1 IsUnit[101X
  
  [29X[2XIsUnit[102X( [[3XR[103X, ][3Xr[103X ) [32X operation
  
  [33X[0;0Y[2XIsUnit[102X returns [9Xtrue[109X if [3Xr[103X is a unit in the ring [3XR[103X, if given, and otherwise in
  its  default  ring (see [2XDefaultRing[102X ([14X56.1-3[114X)). If [3Xr[103X is not a unit then [9Xfalse[109X
  is returned.[133X
  
  [33X[0;0YAn element [3Xr[103X is called a [13Xunit[113X in a ring [3XR[103X, if [3Xr[103X has an inverse in [3XR[103X.[133X
  
  [33X[0;0Y[2XIsUnit[102X may call [2XQuotient[102X ([14X56.1-9[114X).[133X
  
  [1X56.5-2 Units[101X
  
  [29X[2XUnits[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0Y[2XUnits[102X  returns the group of units of the ring [3XR[103X. This may either be returned
  as a list or as a group.[133X
  
  [33X[0;0YAn  element  [22Xr[122X  is called a [13Xunit[113X of a ring [22XR[122X if [22Xr[122X has an inverse in [22XR[122X. It is
  easy to see that the set of units forms a multiplicative group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XUnits( GaussianIntegers );[127X[104X
    [4X[28X[ -1, 1, -E(4), E(4) ][128X[104X
    [4X[25Xgap>[125X [27XUnits( GF( 16 ) );[127X[104X
    [4X[28X<group with 1 generators>[128X[104X
  [4X[32X[104X
  
  [1X56.5-3 IsAssociated[101X
  
  [29X[2XIsAssociated[102X( [[3XR[103X, ][3Xr[103X, [3Xs[103X ) [32X operation
  
  [33X[0;0Y[2XIsAssociated[102X returns [9Xtrue[109X if the two ring elements [3Xr[103X and [3Xs[103X are associated in
  the  ring  [3XR[103X, if given, and otherwise in their default ring (see [2XDefaultRing[102X
  ([14X56.1-3[114X)). If the two elements are not associated then [9Xfalse[109X is returned.[133X
  
  [33X[0;0YTwo  elements [3Xr[103X and [3Xs[103X of a ring [3XR[103X are called [13Xassociated[113X if there is a unit [22Xu[122X
  of [3XR[103X such that [3Xr[103X [22Xu =[122X[3Xs[103X.[133X
  
  [1X56.5-4 Associates[101X
  
  [29X[2XAssociates[102X( [[3XR[103X, ][3Xr[103X ) [32X operation
  
  [33X[0;0Y[2XAssociates[102X  returns  the set of associates of [3Xr[103X in the ring [3XR[103X, if given, and
  otherwise in its default ring (see [2XDefaultRing[102X ([14X56.1-3[114X)).[133X
  
  [33X[0;0YTwo  elements [3Xr[103X and [22Xs[122X of a ring [22XR[122X are called [13Xassociated[113X if there is a unit [22Xu[122X
  of [22XR[122X such that [22X[3Xr[103X u = s[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAssociates( Integers, 2 );[127X[104X
    [4X[28X[ -2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XAssociates( GaussianIntegers, 2 );[127X[104X
    [4X[28X[ -2, 2, -2*E(4), 2*E(4) ][128X[104X
  [4X[32X[104X
  
  [1X56.5-5 StandardAssociate[101X
  
  [29X[2XStandardAssociate[102X( [[3XR[103X, ][3Xr[103X ) [32X operation
  
  [33X[0;0Y[2XStandardAssociate[102X  returns  the  standard associate of the ring element [3Xr[103X in
  the  ring  [3XR[103X,  if  given, and otherwise in its default ring (see [2XDefaultRing[102X
  ([14X56.1-3[114X)).[133X
  
  [33X[0;0YThe  [13Xstandard associate[113X of a ring element [3Xr[103X of [3XR[103X is an associated element of
  [3Xr[103X  which  is,  in  a  ring  dependent  way,  distinguished  among the set of
  associates of [3Xr[103X. For example, in the ring of integers the standard associate
  is the absolute value.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:= Indeterminate( Rationals, "x" );;[127X[104X
    [4X[25Xgap>[125X [27XStandardAssociate( -x^2-x+1 );[127X[104X
    [4X[28Xx^2+x-1[128X[104X
  [4X[32X[104X
  
  [1X56.5-6 StandardAssociateUnit[101X
  
  [29X[2XStandardAssociateUnit[102X( [[3XR[103X, ][3Xr[103X ) [32X operation
  
  [33X[0;0Y[2XStandardAssociateUnit[102X  returns  a  unit  in  the  ring  [3XR[103X such that the ring
  element [3Xr[103X times this unit equals the standard associate of [3Xr[103X in [3XR[103X.[133X
  
  [33X[0;0YIf  [3XR[103X  is not given, the default ring of [3Xr[103X is used instead. (see [2XDefaultRing[102X
  ([14X56.1-3[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xy:= Indeterminate( Rationals, "y" );;[127X[104X
    [4X[25Xgap>[125X [27Xr:= -y^2-y+1;[127X[104X
    [4X[28X-y^2-y+1[128X[104X
    [4X[25Xgap>[125X [27XStandardAssociateUnit( r );[127X[104X
    [4X[28X-1[128X[104X
    [4X[25Xgap>[125X [27XStandardAssociateUnit( r ) * r = StandardAssociate( r );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X56.5-7 IsIrreducibleRingElement[101X
  
  [29X[2XIsIrreducibleRingElement[102X( [[3XR[103X, ][3Xr[103X ) [32X operation
  
  [33X[0;0Y[2XIsIrreducibleRingElement[102X  returns  [9Xtrue[109X if the ring element [3Xr[103X is irreducible
  in  the ring [3XR[103X, if given, and otherwise in its default ring (see [2XDefaultRing[102X
  ([14X56.1-3[114X)). If [3Xr[103X is not irreducible then [9Xfalse[109X is returned.[133X
  
  [33X[0;0YAn  element  [3Xr[103X of a ring [3XR[103X is called [13Xirreducible[113X if [3Xr[103X is not a unit in [3XR[103X and
  if  there  is  no  nontrivial  factorization of [3Xr[103X in [3XR[103X, i.e., if there is no
  representation  of [3Xr[103X as product [22Xs t[122X such that neither [22Xs[122X nor [22Xt[122X is a unit (see
  [2XIsUnit[102X ([14X56.5-1[114X)). Each prime element (see [2XIsPrime[102X ([14X56.5-8[114X)) is irreducible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsIrreducibleRingElement( Integers, 2 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X56.5-8 IsPrime[101X
  
  [29X[2XIsPrime[102X( [[3XR[103X, ][3Xr[103X ) [32X operation
  
  [33X[0;0Y[2XIsPrime[102X  returns  [9Xtrue[109X  if  the  ring element [3Xr[103X is a prime in the ring [3XR[103X, if
  given, and otherwise in its default ring (see [2XDefaultRing[102X ([14X56.1-3[114X)). If [3Xr[103X is
  not a prime then [9Xfalse[109X is returned.[133X
  
  [33X[0;0YAn  element [3Xr[103X of a ring [3XR[103X is called [13Xprime[113X if for each pair [22Xs[122X and [22Xt[122X such that
  [3Xr[103X divides [22Xs t[122X the element [3Xr[103X divides either [22Xs[122X or [22Xt[122X. Note that there are rings
  where  not every irreducible element (see [2XIsIrreducibleRingElement[102X ([14X56.5-7[114X))
  is a prime.[133X
  
  [1X56.5-9 Factors[101X
  
  [29X[2XFactors[102X( [[3XR[103X, ][3Xr[103X ) [32X operation
  
  [33X[0;0Y[2XFactors[102X  returns  the  factorization of the ring element [3Xr[103X in the ring [3XR[103X, if
  given,  and otherwise in in its default ring (see [2XDefaultRing[102X ([14X56.1-3[114X)). The
  factorization  is  returned as a list of primes (see [2XIsPrime[102X ([14X56.5-8[114X)). Each
  element in the list is a standard associate (see [2XStandardAssociate[102X ([14X56.5-5[114X))
  except  the  first  one,  which is multiplied by a unit as necessary to have
  [10XProduct(  Factors(  [3XR[103X[10X,  [3Xr[103X[10X  )  )  = [3Xr[103X[10X[110X. This list is usually also sorted, thus
  smallest prime factors come first. If [3Xr[103X is a unit or zero, [10XFactors( [3XR[103X[10X, [3Xr[103X[10X ) =
  [ [3Xr[103X[10X ][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:= Indeterminate( GF(2), "x" );;[127X[104X
    [4X[25Xgap>[125X [27Xpol:= x^2+x+1;[127X[104X
    [4X[28Xx^2+x+Z(2)^0[128X[104X
    [4X[25Xgap>[125X [27XFactors( pol );[127X[104X
    [4X[28X[ x^2+x+Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27XFactors( PolynomialRing( GF(4) ), pol );[127X[104X
    [4X[28X[ x+Z(2^2), x+Z(2^2)^2 ][128X[104X
  [4X[32X[104X
  
  [1X56.5-10 PadicValuation[101X
  
  [29X[2XPadicValuation[102X( [3Xr[103X, [3Xp[103X ) [32X operation
  
  [33X[0;0Y[2XPadicValuation[102X  is  the  operation to compute the [3Xp[103X-adic valuation of a ring
  element [3Xr[103X.[133X
  
  
  [1X56.6 [33X[0;0YEuclidean Rings[133X[101X
  
  [1X56.6-1 IsEuclideanRing[101X
  
  [29X[2XIsEuclideanRing[102X( [3XR[103X ) [32X Category
  
  [33X[0;0YA  ring  [22XR[122X  is  called  a Euclidean ring if it is an integral ring and there
  exists  a  function  [22Xδ[122X,  called  the  Euclidean  degree, from [22XR-{0_R}[122X to the
  nonnegative  integers,  such that for every pair [22Xr ∈ R[122X and [22Xs ∈ R-{0_R}[122X there
  exists  an  element [22Xq[122X such that either [22Xr - q s = 0_R[122X or [22Xδ(r - q s) < δ( s )[122X.
  In  [5XGAP[105X the Euclidean degree [22Xδ[122X is implicitly built into a ring and cannot be
  changed.  The  existence  of  this  division with remainder implies that the
  Euclidean  algorithm  can be applied to compute a greatest common divisor of
  two elements, which in turn implies that [22XR[122X is a unique factorization ring.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsEuclideanRing( GaussianIntegers );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X56.6-2 EuclideanDegree[101X
  
  [29X[2XEuclideanDegree[102X( [[3XR[103X, ][3Xr[103X ) [32X operation
  
  [33X[0;0Y[2XEuclideanDegree[102X  returns  the  Euclidean degree of the ring element [3Xr[103X in the
  ring  [3XR[103X,  if  given,  and  otherwise  in  its  default ring (see [2XDefaultRing[102X
  ([14X56.1-3[114X)).[133X
  
  [33X[0;0YThe ring [3XR[103X must be a Euclidean ring (see [2XIsEuclideanRing[102X ([14X56.6-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XEuclideanDegree( GaussianIntegers, 3 );[127X[104X
    [4X[28X9[128X[104X
  [4X[32X[104X
  
  [1X56.6-3 EuclideanQuotient[101X
  
  [29X[2XEuclideanQuotient[102X( [[3XR[103X, ][3Xr[103X, [3Xm[103X ) [32X operation
  
  [33X[0;0Y[2XEuclideanQuotient[102X  returns the Euclidean quotient of the ring elements [3Xr[103X and
  [3Xm[103X  in  the  ring  [3XR[103X,  if  given,  and  otherwise  in their default ring (see
  [2XDefaultRing[102X ([14X56.1-3[114X)).[133X
  
  [33X[0;0YThe  ring  [3XR[103X  must  be  a  Euclidean  ring  (see  [2XIsEuclideanRing[102X ([14X56.6-1[114X)),
  otherwise an error is signalled.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XEuclideanQuotient( 8, 3 );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X56.6-4 EuclideanRemainder[101X
  
  [29X[2XEuclideanRemainder[102X( [[3XR[103X, ][3Xr[103X, [3Xm[103X ) [32X operation
  
  [33X[0;0Y[2XEuclideanRemainder[102X  returns  the  Euclidean  remainder of the ring element [3Xr[103X
  modulo  the  ring  element [3Xm[103X in the ring [3XR[103X, if given, and otherwise in their
  default ring (see [2XDefaultRing[102X ([14X56.1-3[114X)).[133X
  
  [33X[0;0YThe  ring  [3XR[103X  must  be  a  Euclidean  ring  (see  [2XIsEuclideanRing[102X ([14X56.6-1[114X)),
  otherwise an error is signalled.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XEuclideanRemainder( 8, 3 );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X56.6-5 QuotientRemainder[101X
  
  [29X[2XQuotientRemainder[102X( [[3XR[103X, ][3Xr[103X, [3Xm[103X ) [32X operation
  
  [33X[0;0Y[2XQuotientRemainder[102X returns the Euclidean quotient and the Euclidean remainder
  of the ring elements [3Xr[103X and [3Xm[103X in the ring [3XR[103X, if given, and otherwise in their
  default  ring  (see  [2XDefaultRing[102X  ([14X56.1-3[114X)).  The  result  is a pair of ring
  elements.[133X
  
  [33X[0;0YThe  ring  [3XR[103X  must  be  a  Euclidean  ring  (see  [2XIsEuclideanRing[102X ([14X56.6-1[114X)),
  otherwise an error is signalled.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQuotientRemainder( GaussianIntegers, 8, 3 );[127X[104X
    [4X[28X[ 3, -1 ][128X[104X
  [4X[32X[104X
  
  
  [1X56.7 [33X[0;0YGcd and Lcm[133X[101X
  
  
  [1X56.7-1 [33X[0;0YGcd[133X[101X
  
  [29X[2XGcd[102X( [[3XR[103X, ][3Xr1[103X, [3Xr2[103X, [3X...[103X ) [32X function
  [29X[2XGcd[102X( [[3XR[103X, ][3Xlist[103X ) [32X function
  
  [33X[0;0Y[2XGcd[102X  returns  the  greatest  common divisor of the ring elements [3Xr1[103X, [3Xr2[103X, [22X...[122X
  resp.  of  the  ring  elements in the list [3Xlist[103X in the ring [3XR[103X, if given, and
  otherwise in their default ring, see [2XDefaultRing[102X ([14X56.1-3[114X).[133X
  
  [33X[0;0Y[2XGcd[102X  returns  the standard associate (see [2XStandardAssociate[102X ([14X56.5-5[114X)) of the
  greatest common divisors.[133X
  
  [33X[0;0YA  divisor of an element [22Xr[122X in the ring [22XR[122X is an element [22Xd∈ R[122X such that [22Xr[122X is a
  multiple  of [22Xd[122X. A common divisor of the elements [22Xr_1, r_2, ...[122X in the ring [22XR[122X
  is  an  element  [22Xd∈  R[122X  which is a divisor of each [22Xr_1, r_2, ...[122X. A greatest
  common  divisor  [22Xd[122X  in  addition  has  the  property that every other common
  divisor of [22Xr_1, r_2, ...[122X is a divisor of [22Xd[122X.[133X
  
  [33X[0;0YNote  that  this in particular implies the following: For the zero element [22Xz[122X
  of [3XR[103X, we have [10XGcd( [3Xr[103X[10X, [110X[22Xz[122X[10X ) = Gcd( [110X[22Xz[122X[10X, [3Xr[103X[10X ) = StandardAssociate( [3Xr[103X[10X )[110X and [10XGcd( [110X[22Xz[122X[10X,
  [110X[22Xz[122X[10X ) = [110X[22Xz[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGcd( Integers, [ 10, 15 ] );[127X[104X
    [4X[28X5[128X[104X
  [4X[32X[104X
  
  [1X56.7-2 GcdOp[101X
  
  [29X[2XGcdOp[102X( [[3XR[103X, ][3Xr[103X, [3Xs[103X ) [32X operation
  
  [33X[0;0Y[2XGcdOp[102X  is  the  operation to compute the greatest common divisor of two ring
  elements [3Xr[103X, [3Xs[103X in the ring [3XR[103X or in their default ring.[133X
  
  
  [1X56.7-3 [33X[0;0YGcdRepresentation[133X[101X
  
  [29X[2XGcdRepresentation[102X( [[3XR[103X, ][3Xr1[103X, [3Xr2[103X, [3X...[103X ) [32X function
  [29X[2XGcdRepresentation[102X( [[3XR[103X, ][3Xlist[103X ) [32X function
  
  [33X[0;0Y[2XGcdRepresentation[102X returns a representation of the greatest common divisor of
  the ring elements [3Xr1[103X, [3Xr2[103X, [22X...[122X resp. of the ring elements in the list [3Xlist[103X in
  the  Euclidean  ring  [3XR[103X,  if given, and otherwise in their default ring, see
  [2XDefaultRing[102X ([14X56.1-3[114X).[133X
  
  [33X[0;0YA representation of the gcd [22Xg[122X of the elements [22Xr_1, r_2, ...[122X of a ring [22XR[122X is a
  list  of ring elements [22Xs_1, s_2, ...[122X of [22XR[122X, such that [22Xg = s_1 r_1 + s_2 r_2 +
  ⋯[122X.  Such  representations  do  not  exist in all rings, but they do exist in
  Euclidean rings (see [2XIsEuclideanRing[102X ([14X56.6-1[114X)), which can be shown using the
  Euclidean algorithm, which in fact can compute those coefficients.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= Indeterminate( Rationals, "a" );;[127X[104X
    [4X[25Xgap>[125X [27XGcdRepresentation( a^2+1, a^3+1 );[127X[104X
    [4X[28X[ -1/2*a^2-1/2*a+1/2, 1/2*a+1/2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[2XGcdex[102X ([14X14.3-5[114X) provides similar functionality over the integers.[133X
  
  [1X56.7-4 GcdRepresentationOp[101X
  
  [29X[2XGcdRepresentationOp[102X( [[3XR[103X, ][3Xr[103X, [3Xs[103X ) [32X operation
  
  [33X[0;0Y[2XGcdRepresentationOp[102X  is  the  operation to compute the representation of the
  greatest common divisor of two ring elements [3Xr[103X, [3Xs[103X in the Euclidean ring [3XR[103X or
  in their default ring, respectively.[133X
  
  [1X56.7-5 ShowGcd[101X
  
  [29X[2XShowGcd[102X( [3Xa[103X, [3Xb[103X ) [32X function
  
  [33X[0;0YThis  function  takes  two elements [3Xa[103X and [3Xb[103X of an Euclidean ring and returns
  their  greatest common divisor. It will print out the steps performed by the
  Euclidean  algorithm, as well as the rearrangement of these steps to express
  the gcd as a ring combination of [3Xa[103X and [3Xb[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XShowGcd(192,42);[127X[104X
    [4X[28X192=4*42 + 24[128X[104X
    [4X[28X42=1*24 + 18[128X[104X
    [4X[28X24=1*18 + 6[128X[104X
    [4X[28X18=3*6 + 0[128X[104X
    [4X[28XThe Gcd is 6[128X[104X
    [4X[28X = 1*24 -1*18[128X[104X
    [4X[28X = -1*42 + 2*24[128X[104X
    [4X[28X = 2*192 -9*42[128X[104X
    [4X[28X6[128X[104X
  [4X[32X[104X
  
  
  [1X56.7-6 [33X[0;0YLcm[133X[101X
  
  [29X[2XLcm[102X( [[3XR[103X, ][3Xr1[103X, [3Xr2[103X, [3X...[103X ) [32X function
  [29X[2XLcm[102X( [[3XR[103X, ][3Xlist[103X ) [32X function
  
  [33X[0;0Y[2XLcm[102X returns the least common multiple of the ring elements [3Xr1[103X, [3Xr2[103X, [22X...[122X resp.
  of the ring elements in the list [3Xlist[103X in the ring [3XR[103X, if given, and otherwise
  in their default ring, see [2XDefaultRing[102X ([14X56.1-3[114X).[133X
  
  [33X[0;0Y[2XLcm[102X  returns  the standard associate (see [2XStandardAssociate[102X ([14X56.5-5[114X)) of the
  least common multiples.[133X
  
  [33X[0;0YA  least  common  multiple of the elements [22Xr_1, r_2, ...[122X of the ring [22XR[122X is an
  element  [22Xm[122X  that is a multiple of [22Xr_1, r_2, ...[122X, and every other multiple of
  these elements is a multiple of [22Xm[122X.[133X
  
  [33X[0;0YNote  that  this in particular implies the following: For the zero element [22Xz[122X
  of [3XR[103X, we have [10XLcm( [3Xr[103X[10X, [110X[22Xz[122X[10X ) = Lcm( [110X[22Xz[122X[10X, [3Xr[103X[10X ) = StandardAssociate( [3Xr[103X[10X )[110X and [10XLcm( [110X[22Xz[122X[10X,
  [110X[22Xz[122X[10X ) = [110X[22Xz[122X.[133X
  
  [1X56.7-7 LcmOp[101X
  
  [29X[2XLcmOp[102X( [[3XR[103X, ][3Xr[103X, [3Xs[103X ) [32X operation
  
  [33X[0;0Y[2XLcmOp[102X  is  the  operation  to  compute the least common multiple of two ring
  elements [3Xr[103X, [3Xs[103X in the ring [3XR[103X or in their default ring, respectively.[133X
  
  [33X[0;0YThe default methods for this uses the equality [22Xlcm( m, n ) = m*n / gcd( m, n
  )[122X (see [2XGcdOp[102X ([14X56.7-2[114X)).[133X
  
  [1X56.7-8 QuotientMod[101X
  
  [29X[2XQuotientMod[102X( [[3XR[103X, ][3Xr[103X, [3Xs[103X, [3Xm[103X ) [32X operation
  
  [33X[0;0Y[2XQuotientMod[102X  returns  the  quotient  of the ring elements [3Xr[103X and [3Xs[103X modulo the
  ring element [3Xm[103X in the ring [3XR[103X, if given, and otherwise in their default ring,
  see [2XDefaultRing[102X ([14X56.1-3[114X).[133X
  
  [33X[0;0Y[3XR[103X   must  be  a  Euclidean  ring  (see  [2XIsEuclideanRing[102X  ([14X56.6-1[114X))  so  that
  [2XEuclideanRemainder[102X ([14X56.6-4[114X) can be applied. If the modular quotient does not
  exist, [9Xfail[109X is returned.[133X
  
  [33X[0;0YThe  quotient  [22Xq[122X  of  [3Xr[103X  and [3Xs[103X modulo [3Xm[103X is an element of [3XR[103X such that [22Xq [3Xs[103X = [3Xr[103X[122X
  modulo  [22Xm[122X,  i.e.,  such  that  [22Xq  [3Xs[103X - [3Xr[103X[122X is divisible by [3Xm[103X in [3XR[103X and that [22Xq[122X is
  either  zero  (if  [3Xr[103X  is  divisible  by  [3Xm[103X)  or the Euclidean degree of [22Xq[122X is
  strictly smaller than the Euclidean degree of [3Xm[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQuotientMod( 7, 2, 3 );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X56.7-9 PowerMod[101X
  
  [29X[2XPowerMod[102X( [[3XR[103X, ][3Xr[103X, [3Xe[103X, [3Xm[103X ) [32X operation
  
  [33X[0;0Y[2XPowerMod[102X  returns  the  [3Xe[103X-th  power  of  the  ring element [3Xr[103X modulo the ring
  element  [3Xm[103X in the ring [3XR[103X, if given, and otherwise in their default ring, see
  [2XDefaultRing[102X ([14X56.1-3[114X). [3Xe[103X must be an integer.[133X
  
  [33X[0;0Y[3XR[103X   must  be  a  Euclidean  ring  (see  [2XIsEuclideanRing[102X  ([14X56.6-1[114X))  so  that
  [2XEuclideanRemainder[102X ([14X56.6-4[114X) can be applied to its elements.[133X
  
  [33X[0;0YIf  [3Xe[103X is positive the result is [3Xr[103X[10X^[110X[3Xe[103X modulo [3Xm[103X. If [3Xe[103X is negative then [2XPowerMod[102X
  first  tries  to  find  the inverse of [3Xr[103X modulo [3Xm[103X, i.e., [22Xi[122X such that [22Xi [3Xr[103X = 1[122X
  modulo  [3Xm[103X.  If  the  inverse  does  not  exist an error is signalled. If the
  inverse does exist [2XPowerMod[102X returns [10XPowerMod( [3XR[103X[10X, [3Xi[103X[10X, -[3Xe[103X[10X, [3Xm[103X[10X )[110X.[133X
  
  [33X[0;0Y[2XPowerMod[102X  reduces  the  intermediate  values modulo [3Xm[103X, improving performance
  drastically when [3Xe[103X is large and [3Xm[103X small.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPowerMod( 12, 100000, 7 );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X56.7-10 InterpolatedPolynomial[101X
  
  [29X[2XInterpolatedPolynomial[102X( [3XR[103X, [3Xx[103X, [3Xy[103X ) [32X operation
  
  [33X[0;0Y[2XInterpolatedPolynomial[102X returns, for given lists [3Xx[103X, [3Xy[103X of elements in a ring [3XR[103X
  of the same length [22Xn[122X, say, the unique polynomial of degree less than [22Xn[122X which
  has value [3Xy[103X[[22Xi[122X] at [3Xx[103X[22X[i][122X, for all [22Xi ∈ { 1, ..., n }[122X. Note that the elements in
  [3Xx[103X must be distinct.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInterpolatedPolynomial( Integers, [ 1, 2, 3 ], [ 5, 7, 0 ] );[127X[104X
    [4X[28X-9/2*x^2+31/2*x-6[128X[104X
  [4X[32X[104X
  
  
  [1X56.8 [33X[0;0YHomomorphisms of Rings[133X[101X
  
  [33X[0;0YA  [13Xring  homomorphism[113X  is a mapping between two rings that respects addition
  and multiplication.[133X
  
  [33X[0;0YCurrently  [5XGAP[105X  supports  ring  homomorphisms  between  finite  rings (using
  straightforward  methods) and ring homomorphisms with additional structures,
  where  source  and  range  are  in  fact  algebras and where also the linear
  structure is respected, see [14X62.10[114X.[133X
  
  [1X56.8-1 RingGeneralMappingByImages[101X
  
  [29X[2XRingGeneralMappingByImages[102X( [3XR[103X, [3XS[103X, [3Xgens[103X, [3Ximgs[103X ) [32X operation
  
  [33X[0;0Yis  a general mapping from the ring [3XA[103X to the ring [3XS[103X. This general mapping is
  defined  by  mapping  the  entries  in  the list [3Xgens[103X (elements of [3XR[103X) to the
  entries  in  the  list  [3Ximgs[103X  (elements  of  [3XS[103X), and taking the additive and
  multiplicative closure.[133X
  
  [33X[0;0Y[3Xgens[103X need not generate [3XR[103X as a ring, and if the specification does not define
  an  additive and multiplicative mapping then the result will be multivalued.
  Hence, in general it is not a mapping.[133X
  
  [1X56.8-2 RingHomomorphismByImages[101X
  
  [29X[2XRingHomomorphismByImages[102X( [3XR[103X, [3XS[103X, [3Xgens[103X, [3Ximgs[103X ) [32X function
  
  [33X[0;0Y[2XRingHomomorphismByImages[102X  returns  the  ring  homomorphism with source [3XR[103X and
  range  [3XS[103X  that is defined by mapping the list [3Xgens[103X of generators of [3XR[103X to the
  list [3Ximgs[103X of images in [3XS[103X.[133X
  
  [33X[0;0YIf  [3Xgens[103X does not generate [3XR[103X or if the homomorphism does not exist (i.e., if
  mapping  the  generators describes only a multi-valued mapping) then [9Xfail[109X is
  returned.[133X
  
  [33X[0;0YOne can avoid the checks by calling [2XRingHomomorphismByImagesNC[102X ([14X56.8-3[114X), and
  one  can  construct  multi-valued  mappings  with [2XRingGeneralMappingByImages[102X
  ([14X56.8-1[114X).[133X
  
  [1X56.8-3 RingHomomorphismByImagesNC[101X
  
  [29X[2XRingHomomorphismByImagesNC[102X( [3XR[103X, [3XS[103X, [3Xgens[103X, [3Ximgs[103X ) [32X operation
  
  [33X[0;0Y[2XRingHomomorphismByImagesNC[102X  is  the operation that is called by the function
  [2XRingHomomorphismByImages[102X   ([14X56.8-2[114X).   Its  methods  may  assume  that  [3Xgens[103X
  generates  [3XR[103X  as  a ring and that the mapping of [3Xgens[103X to [3Ximgs[103X defines a ring
  homomorphism. Results are unpredictable if these conditions do not hold.[133X
  
  [33X[0;0YFor  creating  a  possibly  multi-valued  mapping  from [3XR[103X to [3XS[103X that respects
  addition  and  multiplication,  [2XRingGeneralMappingByImages[102X  ([14X56.8-1[114X)  can be
  used.[133X
  
  [1X56.8-4 NaturalHomomorphismByIdeal[101X
  
  [29X[2XNaturalHomomorphismByIdeal[102X( [3XR[103X, [3XI[103X ) [32X operation
  
  [33X[0;0Yis  the  homomorphism  of  rings provided by the natural projection map of [3XR[103X
  onto  the  quotient ring [3XR[103X/[3XI[103X. This map can be used to take pre-images in the
  original ring from elements in the quotient.[133X
  
